第六屆 ACM ICPC 全國私立大專校院程式競賽. National Contest for Private Universities, Taiwan 2016 競賽題目

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1 第六屆 ACM ICPC 全國私立大專校院程式競賽 National Contest for Private Universities, Taiwan 016 競賽題目

2 Problem 1: Maximum Subsequence Sum Problem (Time Limit: 3 seconds) Given a sequence containing both negative and positive integers. Find the contiguous subsequence with maximum sum. For example, for input {-, 11, -4, 13-5, -}, the answer is 0 (the nd through the 4th). If all numbers are negative, the result is maximum of them, i.e., smallest in terms of absolute value. Input The input begins with a single positive integer T (1 T 0) on a line by itself indicating the number of test cases following, each of them as described below. The first line of each test case contains a single integer N (1 N 10000) that indicates how many numbers are in the sequence. The second line contains N space-separated integers. All these numbers are between and Output For each test case, output one line of three space-separated integers: the maximum sum, the starting position, the ending position of the contiguous subsequence, respectively. If there is more than one such subsequence, output the one that has minimum length. If there is more than one subsequence of minimum length, output the one that occurs first. Sample Input Sample Output

3 Problem: Unlocking Steps Problem Description (Time Limit: 3 seconds) A combination lock is a locking device where a sequence of numbers is used to open the lock. The sequence is entered by rotating discs with inscribed numbers. Each disc is inscribed with following digits, 0, 1,, 3, 4, 5, 6, 7, 8, and 9, and the digit next to 9 is 0. Each disc can be rotated clockwise or counter clockwise. An unlock step is a neighboring digit rotation. For example, rotating one disc from 1 to costs 1 step. Rotating one disc from 0 to 9 also costs 1 step. There may be several digits on a lock. Given an initial number of a combination lock 11677, the minimum number of steps to unlock the lock to the key number is (1 0) + (4 1) + (6 4) + (10 7) + (7 5) = = 11. Input Format The first line is a number n indicating the number of test cases. Each test case has numbers in one line to represent the initial number and the key number of the lock. Output Format For each test case, output the minimum number of steps to unlock the lock. Technical Specification Number n is an integer. 1 n 10. The length of initial number or key number is 9 at most. Example Sample Input: Sample Output:

4 Problem3: Division of Polynomials (Time Limit: 3 seconds) A polynomial may be represented by two sequences, one contains the degrees in descending order of every term in the polynomial (zero-coefficient terms are not included) and the other contains the corresponding coefficients. For example, a polynomial 6x 3 + 5x - 7 is represented as {3,, 0} and {6, 5, -7}. Given two polynomials, print the resultant quotient and remainder polynomials after division of the two polynomials. The divisor is a monic polynomial in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Input The input begins with a single integer T (1 T 100) on a line by itself indicating the number of test cases following, each of them as described below. For each test case, the first line contains each polynomial term s degree (0 d 100) in descending order, the second line contains each term s coefficient (-100 c 100), and the last two lines form the divisor polynomial. All these numbers are integer. Output For each test case, print the paired sequences in separate lines (degree s first, then coefficient s) that represent resultant quotient polynomial, then similarly print the paired sequences that represent resultant remainder polynomial. Notice that either the quotient or the remainder polynomial may be empty (represented as {0} and {0}). Sample Input

5 Sample Output

6 Problem 4: Multiplicative Cipher (Time Limit: 3 seconds) Problem Description The multiplicative cipher is a type of substitution cipher where each letter of the plaintext is substituted with another letter to form the ciphertext. In the multiplicative cipher, encrypting and decrypting letters involved converting them to numbers, multiplying the key, and then converting the new number back to a letter. Your task is to write a program to decryption message using the multiplicative cipher. That is, given the key of the cipher and ciphertext, your program must output the plaintext. The plaintext/ciphertext contained only plus sign (+), uppercase letters (A to Z) and digits (0 to 9). The plaintext/ciphertext has been encrypted/decrypted via the following method. Encryption Convert the plaintext letters to numerical values as follows: + => 0, A => 1, B =>,, Z => 6, 0 => 7, 1 => 8, => 9,, 9 => 36. The encryption function for a single letter is C = k P (mod 37). P means the number corresponding to plaintext letter. C means the number corresponding to ciphertext letter. k is the key of the cipher. The value k must be chosen such that k and 37 are coprime. Convert the numerical values to ciphertext letters as follows: 0 => +, 1 => A, => B,, 6 => Z, 7 => 0, 8 => 1, 9 =>,, 36 => 9. Decryption Convert the ciphertext letters to numerical values as follows: + => 0, A => 1, B =>,, Z => 6, 0 => 7, 1 => 8, => 9,, 9 => 36.

7 The decryption function for a single letter is P = k -1 C (mod 37). k 1 is the modular multiplicative inverse of a modulo 37. That is, k k -1 = 1 (mod 37). Convert the numerical values to plaintext letters as follows: 0 => +, 1 => A, => B,, 6 => Z, 7 => 0, 8 => 1, 9 =>,, 36 => 9. Example with (k, k -1 ) = (5, 3): Plaintext letters: A B C D Plaintext numbers: Ciphertext numbers: Ciphertext letters: Y M A Z 7 V J 8 Input Format The first line of input indicates the number of test cases, T (1 T 100). For each case, the first line of input is the given cipher key, k (1 k 36). The next line of input is the ciphertext which is composed by one line and cannot exceed 70 characters of length. Output Format For each test case, the output should be a single line containing the original plaintext. Example Sample Input: 30 RFX+3VB+JFA+66U 19 DSHH4 Sample Output: HOW+ARE+YOU+007 HAPPY

8 Problem 5: Set Operations On Segments (time limit: seconds) You are given two sets of horizontal line segments on the X-axis. You are to compute the intersection and union of the two sets. In mathematics, a closed interval is defined as all the points between two endpoints and includes the endpoints. The intersection of two segments is defined as the closed interval where the two segments overlap. It follows from this definition that the intersection cannot be a single point; it must either be another segment or nothing. The union of two segments is defined as the smallest closed interval that contains both segments if and only if the two segments overlap or meet at an endpoint. Otherwise, the union results in the two original segments. Graphically, AB BC = ; AC BC = BC A B C D AB BC = AC ; AC BD = AD ; AB CD = AB and CD Input The first line of input is an integer K, K 10, that specifies the number of test cases for the problem. The rest of input is divided into two parts. The first part begins with a series of assignment statements, one statement per line terminated by the endof-line character, of the following format: For example, A= English alphabet=floating-point number The floating-point number is the coordinate of a point denoted by the alphabet. You can be certain there is no space before or after the assignment statement, and note that there is no space before or after the equal sign. This part of input is terminated by a line that contains two equal signs. The second part of input consists of K test cases, each case on two lines, and the test cases are separated again by a line with two equal signs. Each of the two lines specifies a set of segments, with each segment defined by giving its endpoints that are two alphabets from the previous part of input. There is no space within each group of two alphabets, and a blank separates consecutive groups. Each set of segments is terminated by the end-of-line character. You can be sure that segments from the same set do not overlap. However, segments which are joined at endpoints must be treated as a single segment.

9 Output Output is on two lines for each test case. The first line displays the intersection and the second line displays the union of the two sets. Each line consists of segments that result from a set operation. These segments must be listed from left to right, and each segment is specified by a pair of alphabets, with the alphabet of the left endpoint shown first followed by the alphabet of the right endpoint. As in the input, there is no space within a pair of alphabets, and a space separates two consecutive pairs. If the outcome of the intersection of the given sets is nothing, output "empty" on a single line. Sample Input A=.3 a=4.57 B=3.5 b=1.4 C=1.8 D=7.1 f=0.97 E=0. g=10.88 d=6.3 == CB DB gd fe Ab Ba dg == aa dd gd da Sample Output ba Ba dg Ef Cg empty Ag

10 Problem 6: Maximum Zero-Crossing Window (Time Limit: 3 seconds) The zero-crossing rate (ZCR) is the rate of sign-changes along a signal, i.e., the rate at which the signal changes from positive to negative or back. This feature has been used heavily in both speech recognition and music information retrieval, being a key feature to classify percussive sounds. In this problem, you will be given a sequence of signal strengths of length n, say, s 1, s,, and a window length w. The zero-crossing rate s n zcr p for such a window at location p is defined by where I() zcr p w 1 1 w 1 i1 I s s pi1 pi 0 is an indicator function whose value is 1 if and only if its argument is true. Your task is to find the location p whose corresponding window has the maximum zerocrossing rate. If there are more than one such locations, output the one with minimum index. For example, consider the following signal of length 10, and window length 5. The maximum zero-crossing rate window is located at position 1. Input The first line of input indicates the number of test cases T. These is a blank line before the test data of each case. The test data of each case spans several lines. The first line contains two integers n ( n ) and w ( w n ), and the next serval lines

11 contains n white character separated integers representing the signal strengths whose values are within and Output For each test case, output one line of a single integer, which represents the position of the maximum zero-crossing window. If there are more than one such window, output the one with minimum index. Sample Input Sample Output 1

12 Problem 7: Breaking Bad Ransomware (Time Limit: 3 seconds) Problem Description Cryptolocker is a kind of ransomware to lock victims valuable files from access by a strong cryptographic algorithm for data encryption. The scheme used is usually a public system and even the FBI has told the innocent users that the quick way to recover the files is to pay the ransom. However, the government authorities have been commanded to form a task force named: National Cryptolocker Protection Unit (NCPU in short) to release the innocence. They have found a flawed implementation of cryptolocker You are asked to decrypt the files. This implementation is an offline version of crytolocker. That is, the key is stored in the victim s disk using a sorting scheme based on the prime factorization. The scheme called prime factorization sorting (PFS in short) is described as follows. PFS: Numbers in the range of 1 to n must be sorted in terms of the number of factors in their prime factorization. In case of a tie, the smaller number will come first. For example, 0 = **5, so it has 3 numbers in its prime factorization. Similarly 35 = 5*7 has numbers in its prime factorization. Therefore, 35 will come before 0 according to this criterion. Given a key with m numbers and the plain text to be encrypted (a string), divide the plain text into substrings of length m and the characters of each substring are reordered to create the cipher text. The reordering is based on how the sorted key corresponds to the original key. For example, let's assume that the key is " ". The PFS sorted version of the key is " ". The six numbers in the sorted version have indices in the original key. Each substring in the plain text is reordered accordingly. For example, a substring "number" is reordered and encrypted to "bnmeru". Given the encrypted string: bnmeru and the key: , you are to output the decrypted string: number. When there are duplicate numbers in the key, their relative orders stay the same after sorting. For the key above, this means that the indices are and not

13 Let the last substring in the plain text has length k. If k<m, reorder the last substring using just the first k characters in the original key as its key. Input Format The first line is the number of test cases T. The next two lines are the key with m numbers n1 n nm separated with at least one space and the encrypted text. The maximum length of each encrypted string is L. The character of the string is restricted in 0-9A-Za-z. This is repeated for all the test cases. Output Format Each line should be the decrypted text of the encrypted string in a test case. Technical Specification Number T is an integer. 1 T 0. Numbers L is an integer. 1 L 100. Numbers m is an integer. 1 m 0. Number n1,n,,nm are integers. 1 n 0, n 1, n,, n m 3. Example Sample Input: testtest bnmeru edrlubpiekn Sample Output: testtest number redbluepink

14 Problem 8: Multiple shortest routes (Time Limit: 3 seconds) Problem Description There are n buildings, numbered from 0 to n 1, and m unidirectional roads. Without passing through any intermediate building, each road starts at one building and ends at another. Alice, who wants to go from building 0 to a destination building, knows the exact amount of time needed to go through each road. Please help her determine whether she has two or more quickest routes to choose from. Unless she has no routes at all to her destination, please also find the time needed to go along one of her quickest route(s). You may assume that Alice s destination building is not building 0. Input Format The first line is the number of test cases. Each test case is specified as follows: The first three lines are the number n of buildings, Alice s destination (which is a building) and the number m of edges. Each of the next m lines specifies a road by its starting building, its ending building and the amount of time (which is a positive integer) to go through it, in that order. Consecutive numbers in a line are separated by a space. Output Format For each test case, please output two lines. The first line is yes if there are two or more shortest routes to go from building 0 to Alice s destination, and it is no otherwise. If there is at least one route from building 0 to Alice s destination, then the second line is the minimum amount of time needed to go from building 0 to Alice s destination. Otherwise, it is x. Technical Specification The number of test cases is at most 15. n 100 is an integer. The amount of time needed to go through any road is a positive integer less than or equal to 100.

15 Example Sample Input: Sample Output: no 7 no x yes